The wavelength of `H_(alpha)` line in the hydrogen spectrum is found to be `6563Å` in the laboratory. If the velocity of the milky way is `1.05xx10^(6)ms^(-1)`, then the wavelength of `H_(alpha)` line in the spectrum of milky way will be

Options

A. `6457Å`
B. `6586Å`
C. `7123Å`
D. `7349Å`

Answer

To solve the problem, we need to find the wavelength of the H-alpha line in the spectrum of the Milky Way, given its laboratory wavelength and the velocity of the Milky Way. We will use the Doppler effect formula for light, which relates the change in wavelength to the velocity of the source.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength of H-alpha line in the laboratory, \( \lambda_0 = 6563 \, \text{Å} \)
- Velocity of the Milky Way, \( v = 1.05 \times 10^6 \, \text{m/s} \)
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)

2. **Use the Doppler Effect Formula:**
The formula for the change in wavelength due to the Doppler effect is given by:
\[ v = \frac{\Delta \lambda}{\lambda_0} \cdot c \]
where \( \Delta \lambda = \lambda_m - \lambda_0 \) is the change in wavelength, and \( \lambda_m \) is the observed wavelength in the Milky Way.

3. **Rearranging the Formula:**
We can rearrange the formula to find \( \Delta \lambda \):
\[ \Delta \lambda = \frac{v}{c} \cdot \lambda_0 \]

4. **Substituting the Values:**
Substitute the known values into the equation:
\[ \Delta \lambda = \frac{1.05 \times 10^6}{3 \times 10^8} \cdot 6563 \]

5. **Calculating \( \Delta \lambda \):**
First, calculate the fraction:
\[ \frac{1.05 \times 10^6}{3 \times 10^8} = 0.0035 \]
Now multiply this by \( 6563 \):
\[ \Delta \lambda = 0.0035 \cdot 6563 \approx 22.9355 \, \text{Å} \]

6. **Finding the Observed Wavelength:**
Now, we can find the observed wavelength \( \lambda_m \):
\[ \lambda_m = \lambda_0 + \Delta \lambda = 6563 + 22.9355 \approx 6585.9355 \, \text{Å} \]

7. **Rounding the Result:**
Rounding this to the nearest whole number gives:
\[ \lambda_m \approx 6586 \, \text{Å} \]

### Final Answer:
The wavelength of the H-alpha line in the spectrum of the Milky Way is approximately \( 6586 \, \text{Å} \).

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